Алгебра
Английский
Биология
География
Геометрия
История
Русский
Обществознание
Физика
Химия
Информатика
Литература
Математика10. Решите уравнения: 1) sin2x = -1/2; 2) cos x/2 = √2/2; 3) tg 3x = -1; 4) cos²x = sin²x; 5) tg²x + 1/tg²x = 10/3; 6) sin²2x = sin²x; 7) (sinx + cosx)² = 2; 8) sinx + sin 2x = 0; 9) sinx + cos²x = 5/4; 10) sinxcosx = √2/4.
Ответ:
1) sin2x = -1/2;
$$2x = arcsin(-\frac{1}{2}) + 2\pi n, n \in Z$$ или $$2x = \pi - arcsin(-\frac{1}{2}) + 2\pi n, n \in Z$$
$$2x = -\frac{\pi}{6} + 2\pi n, n \in Z$$ или $$2x = \pi + \frac{\pi}{6} + 2\pi n, n \in Z$$
$$x = -\frac{\pi}{12} + \pi n, n \in Z$$ или $$x = \frac{7\pi}{12} + \pi n, n \in Z$$
Ответ: $$x = -\frac{\pi}{12} + \pi n, n \in Z$$ или $$x = \frac{7\pi}{12} + \pi n, n \in Z$$
2) cos(x/2) = √2/2;
$$\frac{x}{2} = arccos(\frac{\sqrt{2}}{2}) + 2\pi n, n \in Z$$ или $$\frac{x}{2} = -arccos(\frac{\sqrt{2}}{2}) + 2\pi n, n \in Z$$
$$\frac{x}{2} = \frac{\pi}{4} + 2\pi n, n \in Z$$ или $$\frac{x}{2} = -\frac{\pi}{4} + 2\pi n, n \in Z$$
$$x = \frac{\pi}{2} + 4\pi n, n \in Z$$ или $$x = -\frac{\pi}{2} + 4\pi n, n \in Z$$
Ответ: $$x = \frac{\pi}{2} + 4\pi n, n \in Z$$ или $$x = -\frac{\pi}{2} + 4\pi n, n \in Z$$
3) tg 3x = -1;
$$3x = arctg(-1) + \pi n, n \in Z$$
$$3x = -\frac{\pi}{4} + \pi n, n \in Z$$
$$x = -\frac{\pi}{12} + \frac{\pi n}{3}, n \in Z$$
Ответ: $$x = -\frac{\pi}{12} + \frac{\pi n}{3}, n \in Z$$
4) cos²x = sin²x;
$$cos^2x - sin^2x = 0$$
$$cos2x = 0$$
$$2x = \frac{\pi}{2} + \pi n, n \in Z$$
$$x = \frac{\pi}{4} + \frac{\pi n}{2}, n \in Z$$
Ответ: $$x = \frac{\pi}{4} + \frac{\pi n}{2}, n \in Z$$
5) tg²x + 1/tg²x = 10/3;
$$tg^2x + ctg^2x = \frac{10}{3}$$
$$tg^2x + \frac{1}{tg^2x} = \frac{10}{3}$$
Пусть $$y = tg^2x$$, тогда $$y + \frac{1}{y} = \frac{10}{3}$$
$$3y^2 + 3 = 10y$$
$$3y^2 - 10y + 3 = 0$$
$$D = 100 - 4 \cdot 3 \cdot 3 = 100 - 36 = 64$$
$$y_1 = \frac{10 + 8}{6} = \frac{18}{6} = 3$$
$$y_2 = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$$
$$tg^2x = 3$$ или $$tg^2x = \frac{1}{3}$$
$$tgx = \sqrt{3}$$ или $$tgx = -\sqrt{3}$$ или $$tgx = \frac{1}{\sqrt{3}}$$ или $$tgx = -\frac{1}{\sqrt{3}}$$
$$x = \frac{\pi}{3} + \pi n, n \in Z$$ или $$x = -\frac{\pi}{3} + \pi n, n \in Z$$ или $$x = \frac{\pi}{6} + \pi n, n \in Z$$ или $$x = -\frac{\pi}{6} + \pi n, n \in Z$$
Ответ: $$x = \pm \frac{\pi}{3} + \pi n, n \in Z$$ или $$x = \pm \frac{\pi}{6} + \pi n, n \in Z$$
6) sin²2x = sin²x;
$$sin^22x - sin^2x = 0$$
$$(sin2x - sinx)(sin2x + sinx) = 0$$
$$sin2x - sinx = 0$$ или $$sin2x + sinx = 0$$
$$2sinxcosx - sinx = 0$$ или $$2sinxcosx + sinx = 0$$
$$sinx(2cosx - 1) = 0$$ или $$sinx(2cosx + 1) = 0$$
$$sinx = 0$$ или $$cosx = \frac{1}{2}$$ или $$cosx = -\frac{1}{2}$$
$$x = \pi n, n \in Z$$ или $$x = \frac{\pi}{3} + 2\pi n, n \in Z$$ или $$x = -\frac{\pi}{3} + 2\pi n, n \in Z$$ или $$x = \frac{2\pi}{3} + 2\pi n, n \in Z$$ или $$x = -\frac{2\pi}{3} + 2\pi n, n \in Z$$
Ответ:$$x = \pi n, n \in Z$$ или $$x = \pm \frac{\pi}{3} + 2\pi n, n \in Z$$ или $$x = \pm \frac{2\pi}{3} + 2\pi n, n \in Z$$
7) (sinx + cosx)² = 2;
$$sin^2x + 2sinxcosx + cos^2x = 2$$
$$1 + 2sinxcosx = 2$$
$$2sinxcosx = 1$$
$$sin2x = 1$$
$$2x = \frac{\pi}{2} + 2\pi n, n \in Z$$
$$x = \frac{\pi}{4} + \pi n, n \in Z$$
Ответ: $$x = \frac{\pi}{4} + \pi n, n \in Z$$
8) sinx + sin 2x = 0;
$$sinx + 2sinxcosx = 0$$
$$sinx(1 + 2cosx) = 0$$
$$sinx = 0$$ или $$cosx = -\frac{1}{2}$$
$$x = \pi n, n \in Z$$ или $$x = \frac{2\pi}{3} + 2\pi n, n \in Z$$ или $$x = -\frac{2\pi}{3} + 2\pi n, n \in Z$$
Ответ: $$x = \pi n, n \in Z$$ или $$x = \pm \frac{2\pi}{3} + 2\pi n, n \in Z$$
9) sinx + cos²x = 5/4;
$$sinx + 1 - sin^2x = \frac{5}{4}$$
$$sin^2x - sinx + \frac{1}{4} = 0$$
$$(sinx - \frac{1}{2})^2 = 0$$
$$sinx = \frac{1}{2}$$
$$x = \frac{\pi}{6} + 2\pi n, n \in Z$$ или $$x = \frac{5\pi}{6} + 2\pi n, n \in Z$$
Ответ: $$x = \frac{\pi}{6} + 2\pi n, n \in Z$$ или $$x = \frac{5\pi}{6} + 2\pi n, n \in Z$$
10) sinxcosx = √2/4.
$$\frac{1}{2}sin2x = \frac{\sqrt{2}}{4}$$
$$sin2x = \frac{\sqrt{2}}{2}$$
$$2x = \frac{\pi}{4} + 2\pi n, n \in Z$$ или $$2x = \frac{3\pi}{4} + 2\pi n, n \in Z$$
$$x = \frac{\pi}{8} + \pi n, n \in Z$$ или $$x = \frac{3\pi}{8} + \pi n, n \in Z$$
Ответ:$$x = \frac{\pi}{8} + \pi n, n \in Z$$ или $$x = \frac{3\pi}{8} + \pi n, n \in Z$$
Похожие
